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Matrix word and elementary row reduction rules

  1. Oct 28, 2008 #1
    1. The problem statement, all variables and given/known data
    a house plan has 3 different floor plans:
    Plan A- 3 three-bedroom units, 7 two-bedroom units, and 8 one-bedroom units
    Plan B- 4 three- bedroom units, 4 two-bedroom units, and 8 one-bedroom units
    Plan C- 5 three-bedroom units, 3 two-bedroom units, and 9 one bedroom units
    is it possible to have 66 three- bedroom units, 74 two-bedroom units, 136 one-bedroom units
    2. Relevant equations
    all elementary row reduction rules

    3. The attempt at a solution
    so far ive put the above into a matrix
    3 7 8|18
    4 4 8|16
    5 3 9|17
    after do all elementary row operations this is my result
    1 -3 0|-2
    0 1 1/2|3/2
    0 0 0|-5
    now i know that if the result in the last row is what it is here it has no solution, but does this mean that the above problem is not possible? is there something more i have to do? and also in my final solution matrix correct?
  2. jcsd
  3. Oct 28, 2008 #2


    Staff: Mentor

    I'm very surprised that you went from a word problem directly to an augmented matrix that supposedly represents a system of equations, apparently skipping the step of producing the system of equations. If you did, you didn't show the system or mention it.

    The augmented matrix you show represents this system:
    3x + 7y + 8z = 18
    4x + 4y + 8z = 16
    5x + 3y + 9z = 17

    What exactly do x, y, and z represent? If this system had been consistent and you had been able to solve it, what would have x, y, and z represented? I don't mean their numeric values.

    Where did you get the constants in the last column of the augmented matrix? Did you just add up the numbers in the row? That's what it looks like.

    One of the questions you asked was whether your final solution matrix correct. A better question would have been, is my initial matrix correct?
  4. Oct 28, 2008 #3


    Staff: Mentor

    As it turns out, there are an infinite number of solutions for the system I'm working with, but only two of them are reasonable. I've checked them both and they give me the right number of one-, two-, and three-bedroom units, so I'm pretty confident I'm on the right track.
  5. Oct 29, 2008 #4


    Staff: Mentor

    Hey, I wasn't trying to scare you away--I was trying to get you thinking before you started mechanically row-reducing an augmented matrix.

    The problem is asking how many plan A floor plans and how many plan B floor plans and how many plan C floor plans can you use to come up with 66 3-BR apts, 74 2-BR apts, and 136 1-BR apts.
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